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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Intermediate normalizing extensions


Authors: A. G. Heinicke and J. C. Robson
Journal: Trans. Amer. Math. Soc. 282 (1984), 645-667
MSC: Primary 16A56; Secondary 16A26, 16A34, 16A55, 16A66
MathSciNet review: 732112
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Abstract: Relationships between the prime ideals of a ring $ R$ and of a normalizing extension $ S$ have been studied by several authors recently. In this work, most of these known results are extended to give relationships between the prime ideals of $ R$ and of $ T$ where $ T$ is a ring with $ R \subset T \subset S$, and $ S$ is a normalizing extension of $ R$: such rings $ T$ are called intermediate normalizing extensions of $ R$.

One result ("Cutting Down") asserts that for any prime ideal $ J$ of $ T$, $ J \cap R$ is the intersection of a finite set of prime ideals $ {P_i}$ of $ R$, uniquely defined by $ J$, whose corresponding factor rings $ R/{P_i}$ are mutually isomorphic. The minimal members of the family of $ {P_i}$'s are the primes of $ R$ minimal over $ J \cap R$, and an "incomparability" theorem is proved which shows that no two comparable primes of $ T$ can have their intersections with $ R$ share a common minimal prime. Other results include versions of the "lying over" and "going up" theorems, proofs that chain conditions such as right Goldie or right Noetherian pass between $ T/J$ and each of the rings $ R/{P_i}$, and a demonstration that the "additivity principle" holds.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0732112-2
Article copyright: © Copyright 1984 American Mathematical Society